CALCULATION OF EXTERNAL DOSE

FROM DISTRIBUTED SOURCE*

CONF-8606139--2

Health Physics Summer School DE86 012617University Park, Pennsylvania

June 23-27. 1986

D. C. Kocher

DISCLAIMER

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Research sponsored by the U.S. Department of Energy under contractDE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.

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DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED

CALCULATION OF EXTERNAL DOSE FROM DISTRIBUTED SOURCES

D. C. KocherHealth and Safety Research D i v i s i o n

Oak Ridge National LaboratoryOak Ridge. TN 37 831

INTRODUCTION

This paper d i s cus se s a r e l a t i v e l y simple c a l c u l a t i o n a l method,c a l l e d the point kernel method (F068), for es t imat ing external dose fromd i s t r i b u t e d sources that emit photon or e l e c t r o n rad ia t ions . Thep r i n c i p l e s of the point kernel method are emphasized, rather than thepresenta t ion of extens ive s e t s of c a l c u l a t i o n s or tables of numericalr e s u l t s . A few c a l c u l a t i o n s are presented for simple source geometriesas i l l u s t r a t i o n s of the method, and r e f e r e n c e s and d e s c r i p t i o n s areprovided for other calculations in the l i terature . This paper alsodescribes exposure situations for which the point kernel method i s notappropriate and other, more complex, methods must be used, but thesemethods are not discussed in any detai l .

Photon sources usually are of greatest interest in externaldosimetry, and the point kernel method has been used to estimateexternal photon dose for a large number of idealized source geometriesthat cay occur either in the workplace or in the environment. Thecalculations for photon sources presented in th i s paper generallyrepresent absorbed dose in a ir at the location of exposed individuals.The conversion from absorbed dose in air to dose equivalents in variousbody organs depends on the specific organ and on the energy anddirectional dependence of the spectrum of photons in air at the receptorlocation. The photon f i e ld at the receptor position depends in turn onthe source geometry, the spectrum of photons emitted from the source,and the scattering and absorption of emitted photons between the sourceand receptor locations. For example, rat ios of organ doses to dose inair as a function of emitted photon energy for immersion in a seai-infinite atmospheric cloud are provided in the DObFACTER computer code(KoSla), and these ratios often are applied to other exposure situationseven though the accuracy of the resulting organ doses generally is notknown. Ratios of organ doses to dose in a ir t lso have been calculatedfor unidirectional radiation f ie lds (Bu85; Dr85) and a f in i te Gaussianplume in the atmosphere (Go82).

By acceptance of this article, thepublisher or recipient acknowledgesthe U.S. Gouarnmer.ij right i tretain £ nonexclusive, royalty-freelicense In and to any copyrightcovering the article.

This paper also descr ibes appl icat ion of the point kernel method to

the ca lcu la t ion of external electron dose from d i s t r ibu ted sources. The

pa r t i cu l a r source geometries considered include a semi- inf in i te volume,

an i n f i n i t e plane, and a uniform d i s t r i b u t i o n on the surface of the

skin. The ca lcu la t ions represent dose equivalent to r ad iosens i t ive

t i s sues near the body surface ( e . g . , skin, gonads, and lens of the eye)

ra ther than absorbed dose in a i r . because standard r a t i o s for converting

dose in a i r to doseo in body t i ssues cannot be developed for rad ia t ions

tha t have a f i n i t e range in matter; i . e . , ca lcu la t ions of dose to body

t i s sues must be performed for each p a r t i c u l a r source geometry and

receptor loca t ion .

The quant i ty that is obtained d i r ec t ly from the point kernel method

is the dose r a t e as a function of time t, which we denote by dD/dt.

Given the dose rate as a function of time, the dose received, denoted by

D, then i s the time in tegra l of the dose r a t e :

D = / [dD(t) /dt] dt . (1)

POINT KERNEL METHOD

The point kernel method for estimating external dose r a t e (or any

other response function} a t a receptor loca t ion i s based on the

pr incip le tha t a l l d i s t r i b u t e d sources may be considered as composed of

d i f f e ren t i a l point sources t ha t are i j o t rop i c . The term " i s o t r o p i c "

means t ha t each d i f fe ren t ia l point source emits rad ia t ions uniformly in

a l l d i r e c t i o n s . Thus, the dose rate from a d i s t r ibu ted source may be

obtained by smuming or in tegra t ing the dose r a t e s from the d i f f e r e n t i a l

point sources that comprise the d i s t r ibu ted source. Symbolically, the

point kernel method for est imating the external dose ra te then i s

represented by the equation

dD(r)/dt = / S ( r ' ) K(.r/,_r) d_r' . (2)

where the vector x denotes the location of the veneptor r e l a t i v e to the

origin of the coordinate system, S(j:') djr' i s the source s t rength in

number of photons or e lec t rons emitted per un i t time in tlie spa t ia l

element ax' a t tha locat ion in the source region r e l a t i ve to the or ig in

of the coordinate system denoted by the vector x', K(_r',_r) i s the point

kernel which, in t h i s case, i s the dose at the receptor loca t ion x due

to a uni t point source located at x', and the in tegra t ion ex t en t s over

ths en t i r e source region. The point kernel K depends on the r ad ia t ion

type ( i . e . , photons or electrons), the emitted energy, and the

composition of materials between the source and receptor locations. The

ime dependence cf the dose rate, which is not displayed explicitly in

i.q. (2), resTu..^ from the time dependence of the source strength; the

point kernel itself does not depend on time.

It is important to note that the point kernel K in eq. (2) is

assumed to depend only on the composition of materials along the direct

path between the source and receptor locations; i . e . , the point kernel

depends only on the vector _r - T'. In general, however, the dose rate

at the receptor location also depends on the composition of materials in

regions that are not along the direct path between the source and

receptor, due to the scattering of radiations away from the direct path

that then are re-scattered in some other region and eventually reach the

receptor location. As discussed in the following section, the point

kernel method does take into account the re-scattering of scattered

radiations to the receptor location, but the method also assumes that

the composition of materials for any path of scattered and re—scattered

radiations is the same as the composition along the direct path. If

this assumption is not approximately correct, then the point kernel

method may be inappropriate for estimating external dose.

As displayed in eq. (2), the point kernel method has two distinct

aspects. The first , and generally the more difficult, is the

specification of the point kernel K(r/,_r) for photon or electron

radiations, i . e . , the expression of the point kernel in some functional

or numerical form. This aspect contains all of the "physics" of the

problem, and is considered in the next section of this paper. The

second aspect is the performance of the integral of the point kernel

over the spatial distribution of source strength. This aspect contains

all of the "mathematics" of the problem. For photon sources, the

integration in eq. (2) can, in some cases, be performed in closed form

( i . e . , the result can be written as an equation), and examples of such

cases are presented in this paper. For most photon source geometries

and for estimating tisrue doses from all electron sources, however,

eq. (2) cannot be integrated in closed form, and the solution must be

obtained using numerical methods.

For convenience in evaluating eq. (2), we assume that the receptoris located at the origin of the coordinate system. Then, eq. (2) may bewritten as

dD/dt = / S(r') K(r') d.r' . (3)

where the vector jr' denotes the locat ion of each d i f f erent ia l pointsource and i s the dummy variable of integrat ion. This i s the form ofthe point kernel method that i s user1, i••• the remainder of t h i s paper.

POINT KERNEL FOR PHOTONS AND ELECTRONS

forcul ar form of the equation for ea<

radiation type then i s discussed.

T i i s sec t ion presents a general equation for the point kernel fcphotons and e lectrons . The particular form of the equation for each

General Form of the Point Kernel

For a monoenergetic point source of energy E, the point kernel at areceptor distance r can be expressed in terms of the po in t - i so trop icspec i f ic absorbed fract ion (Be6 8a; L068) , which we denote by £>. Thespeci f ic absorbed fract ion depends on the radiat ion type and the mediumin which the emitted energy i s being absorbed, and i s defined asfol lows:

JKr.E) = fraction of emitted energy E absorbed per gram of materialat distance r from an i sotropic point source.

The point kernel is given in terms of the s p e c i f i c absorbed frac t ion by(Be6 8a)

K(r) = kE D(r,E) , (4)

where

k = 1.6 x 10~1 3 kg-Gy/MeV,E = emitted energy in MeV, andft = s p e c i f i c absorbed fract ion in kg""*.

The constant k i s determined by the de f in i t i on of absorbed dose in grayas one j o u l e of absorbed energy per kilogram of material and theconversion factor from MeV to joules (1.6 x 10~ 1 3 J/MeV). Thus, thepoint kernel K has units of absorbed dose (Gy), and the integral ineq. (2) or (3) has units of absorbed dose rate (Gy/s, or a moreconvenient time basis such as hours or y e a r s ) .

For a point source embedded in an infinite absorbing medium, all of

the emitted energy must be absorbed somewhere within the mediwi. Thus,

the specific absorbed fraction obeys the following important

normalization condition:

4it /£ D(r,E) r2 dr = 1/p , (5)

where p is the density of the medium. This equation holds for any

energy E and any functional form for the specific absorbed fraction, and

is str ict ly a consequence of the principle of conservation of energy.

Specific At sorbed Fraction for Photons

For photons (y), the specific absorbed fraction at distance r from

a point source can be written as the product of four factors (Be6 8a):

$Y(r.EY) = (|ien/p) (l/47rr2) Ben(|ir) esp(-ur) , (6)

where

Hen/p = mass energy-absorption coefficient in medium at location ofof receptor in m^/kg,

Ben = energy-absorption buildup factor in medium between sourceand receptor, and

u = linear attenuation coefficient in medium between source and

receptor in m .

The quantity l/4nr is the geometrical reduction term and gives the

dependence of the dose rate due only to the distance from the point

source; i .e. , in the absence of any scattering or absorption of the

emitted radiations, the number of photons per unit time passing through

ths surface of any sphere centered about the point source must be

conserved. The attenuation term exp(-ur) describes the reduction in the

number of unscattered photons along the direct path between the source

and receptor locations due to scattering or absorption in the medium

between t te two locations; i . e . , this term gives the probability that an

emitted photon will travel between the source and receptor locations

without collision, and the product of the geometrical reduction term and

the attenuation term thus is proportional to the uncoilided flux of

photons reaching the receptor location. The term B i s the buildup

factor (Ch6 8) which takes into account those photons that scatter from

the direct path between the source and receptor locations but eventually

re-scatter back to the receptor. The buildup factor is always greater

than unity, and converts the uncoilided flux to the tota1 flux of

uncoilided and collided photons. Finally, the term He,,/^ describes

energy absorption at the receptor position, and thus is ^;ed to convert

the flux of photons at the receptor location to absorbed dose.

The mass energy-absorption coefficient, linear attenuation

coefficient, and energy-absorption buildup factor all depend on the

emitted photon energy, and tabulations of these quantities in different

media are available in the l i terature [e.g. , see (Tr66), (Ch68), and

(USHEW70)]. Empirical formulas have been developed to approximate

buildup factors for point sources [e.g., see (Tr66) and (Ch68)]. These

formulas usually express the buildup factor as the sum of polynomial or

exponential terms. In the author's experience, a useful approximation

to the buildup factor is Berger's formula (Tr66; Ch6 8):

B(jir) = 1 + (C|ir) exp(Dtir) , (7)

where the coefficients C and D are functions of the emitted photon

energy (D must be less than unity), and are obtained from linear least-

squares f i t s of eq. (7) to tabulated buildup factors [e.g., see (Tr66) ,

(CJi6 8), and (KoSla)]. Berger's formula is useful because it is

reasonably accurate and often leads to an integral of the point kernel

over distributed sources in eq. (3) that can be evaluated in closed form

or in terms of well known integrals.

If more than one medium exists between the source and receptor

locations, then the exponential attenuation term and the buildup factor

in the specific absorbed fraction in eq. (6) must be modified

appropriately. If the distance r from source to receptor consists of n

layers and the ith layer has thickness x t and linear attenuation

coefficient u^, then the exponential attenuation term becomes

+ . . . + nnxn)] • (3)

A useful empirical equation for the buildup factor in a two-layer mediumhas been discussed by Trubey (Tr70). In general, however, buildupfactors for multiple layers are unknown, and simplifications that oftenare used to overcome the absence of reliable calculations are discussedby Chilton (Ch6 8).

In summary, eqs. (4) and (6) and an approximation for the buildup

factor, such as Berger's formula in eq. (7), provide an expression for

the point kernel for photons that can be used in eq. (3) to calculate

external photon dose rates from distributed sources. Values of the

energy-dependent parameters in eqs. (6) and (7) are available in the

literature. The calculation of external dose from distributed photon

sources thus i s reduced to the mathematical problem of performing an

integration over the source region, either numerically or by means of a

closed-form solution. Examples of such calculations are presented later

in this paper.

Specific Absorbed Fraction for Electrons

For electrons, an equation for the specific absorbed fraction,

analogous to eq. (6) for photons, has not been developed. An empirical

equation for calculating external dose from electrons was developed by

Loevinger £t _al. (Lo56) from measurements of the distribution of dose in

air around point sources of beta radiation, and the Loevinger equation

has continued to be used widely [e.g., see (He68) and (He84)] even

though two important factors limit i ts validity. First, since the

equation was derived from measurements of dose from continuous spectra

of electrons in beta decay, the equation does not apply to discrete

Auger and internal conversion electrons that often occur in radioactive

decay. Second, the empirical equation provides a reasonable

approximation to dose from beta radiations only when the distance from

the source to the receptor is comparable to or less than the electron

range for the average energy in the spectrum; i . e . , the equation does

not give an adequate description of dose from beta radiations when only

the highest energy electrons near the endpoint of the spectrum have

sufficient range to reach the receptor location.

In the absence of a generally applicable equation for the specific

absorbed fraction for monoenergetic electrons, Berger (Be73; Be74) has

obtained values numerically from Monte Carlo calculations of electron

transport in water. In order to minimize the dependence of the specific

absorbed fraction en electron energy and, thus, to facilitate

interpolation of tabulated values, Berger has introduced the

dimensionless scaled point kernel, denoted by F(r/r ,E), which is

defined in terms of the specific absorbed fraction for electrons (E) of

energy E by the equation

F ( r / r o' E e ) d ( r / r0

) = 47tP V r ' V f 2 d r ' ( 9 )

where r0 is the mean electron range for energy E in the medium with

density p. Thus, "scaling" of the point kernel is accomplished by

expressing distances in the dimension!ess unit of number of electronranges.

Berger (Be73) har provided extensive tabulations oi electron scaledpoint kernels in water as a function of emitted energy and scaled

distance r/r . water provides a reasonable approximation to tissue-equivalent material. Berger (Be74) a.?, so has developed the followingapproximation for obtaining scaled poirt kernels in air (a) from valuesin water (w):

Fa(u.E ) = oFw(ou.E ) , (10)G 6

where u i s the scaled distance r/r and a i s a scaling parameter that isnearly constant at the value 1.02 for emitted electron energies from0.02 to 2 MeV.

Calculations of electron scaled point kernels or specif ic absorbedfractions have not been performed for more than one medium between thesource and receptor. The problem of transport through two media [ e . g . ,air followed by tissue (water)] has been treated by replacing thedistance between source and receptor by an equivalent thickness of oneof the media and using the scaled point kernel for that medium alone(Eo81a). For example, if the media between source and receptor consistof thicknesses x^ Of densi t ies pj and electron mass stopping powers R ,̂where i = 1 or 2 , then the equivalent distance r^ ' from source toreceptor for the f irst medium alone is approximated by

V = xj. + (R 2 /R 1 ) (p 2 /Pi) i2 • ( 1 1 )

A similar equation i s obtained for the equivalent thickness for thesacond medium alone by reversing the subscripts 1 and 2. For greatestaccuracy, the equivalent thickness should be calculated for the mediumwhich provides the greater shielding.

In summary, electron specif ic absorbed fractions have beenexpressed in terms of the scaled point kernel in eq. (9 ) , and the latteri s available only in numerical tabulations. Thus, the dose rate fromdistributed electron sources in eq. (3) generally can be calculated onlyby numerical integrations. Application of the electron scaled pointkernel to the calculation of external dose for particular sourceconfigurations i s i l lustrated later in this paper.

APPLICATION TO SPECTRA OF PHOTONS ANDELECTRONS IN RADIOACTIVE

The equations developed in t h i s paper ap>J/ to monoenergeticsources of photons or e lec t rons . In general , however, dose ca-lcnl ationsare concerned with exposure to radionucl ides which emit a spectrum ofphotons and e lect rons .

The spectrum of photons emitted by radionuclides usual ly is assumedto consist en t i re ly of d i s c r e t e y- and X-rays, although the continuousspectrum of bremsstrahlung can be important for radionuclides that arepure beta emit ters [ e . g . , see (DiSO) for methods of incorporatingbremsstrahlung in external dose c a l c u l a t i o n s ] . If A denotes theac t iv i ty of a radionucl ide and D (£) denotes the dose from amonoenergetic photon source of unit a c t i v i t y , then the dose from thediscre te spectrum of photons emitted by the radionuclide i s given by

DT = A | f i y D v ( E i Y ) . (12)

where f. j s the in tens i ty of the ith photon of energy E^ in number perdecay and the summation includes a l l d i s c r e t e photons in the decayspectrum.

The spectrum of e lec t rons emitted by radionuclides may consist ofd iscre te Auger and in te rna l conversion e lec t rons and continuouselec t rons from beta decay. If A again denotes the ac t iv i ty of aradionuclide and D (E) denotes the dose from a monoenergetic electronsource of un i t ac t iv i ty , then the dose from the d iscre te and continuousspectrum of electrons emitted by the radionuclide is given by

De = A [ | f ie V E i e > +l f j P ft NJp(E) De(E) dE] , (13)

where £ ̂ and fjg are the i n t e n s i t i e s in number per decay of the i thdiscre te electron (e) with energy E i e a n d of electrons from the j t hcontinuous beta t r ans i t ion (p) with endpoint energy EJm, respect ively ,aD<* ^ j p ^ ' * s *^e probabi l i ty per unit energy for emission of anelect ron with energy between E and E + dE for the j t h continuous betat r ans i t i on as obtained from the Fermi theory of beta decay (Y/u66).

10

DOSE RATES FROM DISTMBFTED SOURCES

Tiiir sect ion presents exwnp.; e ca lcula t ions of dose r a t e s fromdis t r ibu ted sources of photons •_- e lec t ron; s? ' ' l u s t r a t i on* c : •"--point kernel method. Discussions of other ca lcu la t ions tha t areavai lable in the l i t e r a t u r e also are included; these discussionspa r t i cu la r ly emphasize ca lcu la t ion of external photon dose from sourcesin the form of a f i n i t e Gaussian plume in the atmosphere, since t h i s i san important problem in environmental dose assessments.

In f in i t e or Semi-Infinite Volume Source of Photons or Electrons

A source region that i s assumed to be an i n f i n i t e or a semi-inf ini te volume with uniform source concentration often i s used inenvironmental dose assessments to approximate a f i n i t e source regionwhen the contaminated medium is a i r or water and an exposed individualis immersed in the medium. Estimation of the absorbed dose ra te frominf in i t e , uniformly contaminated volume sources of photons or e lec t ronsi s pa r t i cu la r ly simple, because of the normalization condition on thespecific absorbed fract ion given in eq. (5) ; i . e . , the absorbed dosera te in the median i t s e l f can be evaluated without exp l i c i t knowledge ofthe specific absorbed f ract ion. For monoenergetic photon (?) orelectron (e) sources of energy E and concentrat ion S in an i n f i n i t emediinn, the absorbed dose ra te at any point in the source region i sobtained from eqs. (3)-(5) as

(dD/dt)y = syfcEy/p , (14)

(dD/dt) e = SekE£/p , (15)

where p is the density of the medium.

For a semi- inf ini te source region, the expossd individual usual lyis assumed to be located at the boundary of the region (e.g. , anindividual immersed in a semi- inf ini te atmospheric cloud while standingon the ground surface) . In t h i s example, to a good approximation, thedose rate in a i r at the boundary of the medium would be one-hal f of thevalues obtained from eqs. (14) or (15). However, in ca lcula t ing organdoses to exposed individuals , i t i s customary to assume that the pointa t which the dose is calculated i s 1 m above the ground surface, ra therthan at the boundary of the source region. In t h i s case, the reduct ionin dose ra te due to the medium being semi- inf in i te i s not p rec i se ly

11

one-half for photons, and the deviations from this value can be as muchas a factor of two at low energies (Ry81; Ja85). For rloctrons, thedose rate in air at 1 m above the 3~ound surface is ihe same as the doserate in an infinite atmospheric \,. oud if the electron i-nge in air islsss than 1 m, but is reduced by a factor between one-half and one ifthe range in air is greater than 1 m.

As mentioned in the Introduction, the conversion from absorbed dosexn air for photons to dose equivalent to various body organs forimmersion in an infinite or semi-infinite medium is accomplished usingstandard conversion factors that are available in the literature(KoSla). For electrons, Berger (Be74) hes shown that the dose rate atdepth x in tissue for immersion in an infinite water (w) medium can bewritten as

(dD/dt)B = (SekEe/pw) q(x.EE) G(x/r^.EE) . (16)

where G, which is analogous to the organ-dose to air-dose ratio forphotons, is called the geometrical reduction factor and is defined asthe ratio of the dose rate at depth x in a semi-inf ini te tissue mediumto the dose rate in. a sem i-inf ini te water medium, rw j . s the meanelectron range in water for energy E, and q is a leakage correctionfactor that accounts for those electrots that backscatter from tissueinto the source region. In terms of the scaled point kernel F definedin eq. (9), the geometrical reduction factor is given by (Be7 4)

G U / r o ' E E ) = ( 1 / 2 ) -C/rw [ 1 " U/TZ)M FW(*.Ee) du , (17)o

where the factor one-half takes into account that for an individualimmersed in an infinite source medium, electrons from only half themedium, can penetrate the body and irradiate the tissues of interest.Values of the geometrical reduction factor and the leakage correctionfactor have been tabulated by Berjer (Be73; Ee7 4). Thus, thecalculation of electron dose rates in tissue from immersion in aninfinite water medium reduces to a problem of interpolating with energyand depth in tissue from tabulated values of the parameters G and q ine q . ,16).

For immersion in contaminated a i r ( a ) , the e lec t ron dose rate int i s sue is similar to eq. (16):

(dD/dt)e = (SekEE/pa) (R*/a) q(x.E£) G(x/r*,Ee) . (18)

where R/a i s the r a t i o of energy absorption in t i s sue to energy

12

absorption in air for emitted energy E (Be74; Ko81a) and q and G are

defined with eq* (16).

In env iror"~'ital dose assessments, i t often is useful .. > calculatea quantity call.ed the dose-rate conversion factor, which is defined asthe external dose rate per unit concentration of a radionuclide. Thesefactors embody all aspects of the calculation of external dose forparticular exposure situations except the radionuclide concentrations.Photon and electron dose-rate conversion factors for many radionuclideshave been tabulated for immersion in a semi-infinite atmospheric cloudand an infinite water medium [e.g. , see (Eo81a), (Ko81b), and (Eo83)].

Infinite Line. Plane, or Slab Source of Photons or Electrons

Other idealized source geometries that commonly are used in dose

assessments and that lead to tractable solutions for the point kernel

method include an infinite and uniformly contaminated line, plane, or

slab source. These geometries may be regarded as linear sources in one,

two, or three dimensions.

Line Source. From eqs. (3) and (4), the dose rate from photons or

electrons at a perpendicular distance z from an infinite line source of

uniform source concentration, is given by

dD/dt = (kE)S F, D(r,E) dr . (19)

For photon sources, the specific absorbed fraction is obtained fromeq. (6); and, if we use the buildup factor from eq. (7), we obtain thefollowing equation for the dose rate:

(dD/dt) = K r (1/r2) [1 + (Cur) exp(Dnr)] exp(-tir) dr

= K { / " (1/r2) exp(-nr) dr \:'

+ Cn rz ( l / r ) e2p[(D-l)ur] dr } , (20)

where K̂ i s a n energy-dependent parameter given by

\ = (kEY)Sy(,ien/p>/4n . (21)

Neither of the integrals in eq. (20) can be evaluated in closed fora.

Ilowever, in terms of the well known first-order exponential integral,E j , given by

13

exp(-ar) dr . (22)

where a is any constant, the solution o eq. (20) can be written in the

following form:

{(1/z) + (C|i)E1[ (D-l)fiz) ]} . (23)

Tlie exponential integrals m eq. (23) can be evaluated using polynomial

and rational approximations given by eqs. 5.1.53 and 5.1.54 of Gautschi

and Cahill (Ga65). As will be evidenced by the solutions for infinite

plane and slab sources presented below, the solutions of the point

kernel method for linear source geometries involve exponential integrals

when tlie Berger form of the buildup factor is used.

For electron sources, we use the scaled point kernel in eq. (9),

and obtain the dose rate at the receptor location in the following form:

(dD/dt)g = d(r/rQ)

where

= (kEe)Sg/4np .

If we define the dummy variable of integration as

= r / r

0 ,

then we obtain the solution in the form

(24)

(25)

(26)

(dD/dt)e = (K2/rQ2) F(u,Eg) du . (27)

As in all calculations of dose rates from electrons, eq. (27) must be

evaluated numerically using tabul ated values of the scaled point kernel

F(u, E )̂ (Be73). We also note that the integrand in eq, (27) has the

same functional form as the integrand in the first term in eq. (20) for

the photon dose rate, if we replace the exponential attenuation term for

photons by the scaled point kernel for electrons.

Plane Source. From eqs. (3) and (4), the dose rate from photons orelectrons from an infinite plane source of uniform source concentrationcan be written in the form

dD/dt = (kE)S /( 2 8 )

14

where a denotes the plane surface. The element of surface area, da, on

the plane surface is given by-

da = 2nr dr . (29)

For photon sources, if we again use t \e specific absorbed fraction

from eq. (6) and the buildup factor from eq. (7) , then the dose rate at

a perpendicular distance z from the plane source is given by the

following equation:

(dD/dt) = K a f (1/r) [1 + (Cjir) exp(D|ir)] exp(-nr) dr

= K, { f° (1/r) exp(-nr) dr

+ Cn r exp[(D-l)|ir] dr } , (30)

where

K3 = (kEv)SY((ien/p)/2 . (31)

The first term in eq. (30) is just the first-order exponential integral

in eq. (22), and the second term can be integrated in closed form.

Thus, the dose rate becomes

(dD/dt)^ = £3 {El(jiz) + [C/(D-1>] exp[(D-l);iz]} • (32)

Evaluation of photon dose rates from a contaminated ground surfaceis a particularly common problem in environmental dose assessments.Dose rates per unit concentration of radionuclides on an infinite anduniformly contaminated ground surface have been tabulated in theliterature (Ko81a; Ko83).

For electron sources, we again use the scaled point kernel in

eq. (9). Using the element of surface area da in eq. (29), we obtain

the dose rate at the receptor location in the following form:

(dD/dt)e = (K4/rQ) r z / t <l/»> F(u.E ) du . (33)o

where

K4 = (k£g)S6/p , (34)

and we have again used the dummy variable of integration u in eq. (26).

We note the similarity between the integrand in eq. (33) for electrons

15

and the integrand in the f i rs t term of eq. (30) for photon.':. Again,

eq. (33) must be ?-*aluated by numerical integrations using tabulated

values of the (-1^ ^rsn scaled point kernel (Be73).

The electron cose rate from an infinite, uniformly contaminated

plane source in eq. (33). based on the electron scaled point kernels cf

Berger (Be73; Be74), is applicable to two important exposure situations.

The first is exposure of an individual standing on a contaminated ground

surface; and dose rates to skin per unit concentration of radionuclides

on the ground have been tabulated in vhe l i terature (Ko81a; Ko31b). The

second is exposure from radionucl ides deposited on the surface of the

skin; and dose rates to skin per unit concentration of radionuclides on

the body surface also have been calculated (He72: Ko85e).

Slab Source. For an infinite, uniformly contaminated slab source,we consider only the dose rate from photons; for electrons, the range ofthe emitted particles in the source volume generally will be so shortthat the source is effectively a place, We also assume in hesecalculations that the scattering and absorption of the (jmitteu photonsin the medium between source and r-uer+o.-: locations ( i . e . , air) can beneglected compared with scattering •»•;« absorption in the source mediumitself. This is a good approximation, for example, for an individualstanding on the ground surface exposed to a uniformly contaminated layerof soil, With this approximation, the photon dose rate near the slabsource will depend on the thickness of tae slab, but will not depend onthe perpendicular distance between the source and receptor.

The photon dose rate from an infinite, uniformly contaminated slab

source is most easily obtained by integrating the dose rate from an

infinite plane source over the thickness of the slab. Thus, if x is the

thickness of the slab, ug denotes the linear attenuation coefficient of

photons in the source region, and C$ a n d D s denote the coefficients for

the Berger form of the buildup factor in eq. (7) in the source region,

then the dose rate for a slab source can be obtained from the dose rate

for a plane source in eq. (32) as

(dD/dt)r = K3 / * (E^^z ) + [CS/(DS-1)] eXp[(Ds-l)ns2]) dz . (35)

Using the known recursion relation for derivatives of exponentialintegrals given by (Ga65)

dEn(az)/d(az) = -En_1(az) . (36)

where a is again any constant and n is any integer greater than 1, and

16

the equation for the second-order exponential iategral, E^, given by

(Gab 5)

E2(a3> = exp(-az) - (az)E1(az) , (37)

the result ist

(dD/dt)y = (K3/n3) {1 - Ejdigx)

+ lC s/(D s-l)2] tl " e?p[(Ds-l)^8x]}} . (38)

For the general case of a uniform slab source with upper and lower

boundaries x^ and x2 with x^ > 0, the dose rate is given by

<dD/dt)y = (K3/ns) {E2(n3Xl) - E2(,isx2) + [CS/(DS-1)23

s-l)n sx2]}} • (39)

Pose rates per unit concentration of monoenergetic photon emitters

uniformly distributed in slab sources of varying thickness in soil have

been tabulated in the l i terature (Ko85b)- These results have been

applied to the calculation of dose rates per unit concentration of

radionucl iaes in slab sources at various distances below the ground

surface (Sj84).*

Gaussian Plume _in the Atmosphere

The Gaussian plume model [e.g., see (Gi68) and (Ha82)] i s widelyused to predict atmospheric transport of radionuclides for both routine(chroaic) and accidental (acute) releases. Thus, the estimation ofexternal dese from exposure to an atmospheric cloud of radionuclides inthe form of a Gaussian plume, and to ridionuclides deposited on theground surface from such a plume, is an important problem inenvironmental dose assessments.

The simplest and most straightforward method for estimating

external dose from an atmospheric clume and from radioactivity deposited

from the plume onto the ground surface is to v»se the Gaussian plume

model and models for wet and dry deposition [e.g. , see (Ha82), (Se84),

and (S134)] to calculate concentrations of radionuclides in the air at

* The dose rates per unit concentration of Sj orten .et _§_1. (Sj84)should be increased by a factor of 920 to correct an error in thetabulation.

17

ground level and on the ground surface at any location of interest, and

then t i calculate external dose rates at those locations by assuming

that fh> estimated concentrations are uniformly distributed in a ser*" •

infinite atmospheric clo-ad or on an infinite ground plane. This metLod

thus avoids the problem of performing numerical integrations of the

point kernel over the finite Gaussian distributions of activity, and

embodies the approaches to calculating external dose developed in this

paper, i . e . , eq. (14) modified by a factor of one-half for a semi-

infinite atmospheric cloud and eq. (32) for an infinite plane surface,

for which tabulations of dose rates per unit concentration of

radionncl ides are available in the li terature (Ko81a; Ko81b; Ko83).

The calculational method involving a Gaussian plume model and

dose-rate conversion factors for a semi-infinite cloud and an infinite

plane surface is a common feature of computer codes that perform routine

dose assessments for atmospheric releases of radioactivity, and include

consideration of several exposure pathways and large numbers of

radionucl ides [e.g., see (Mo79)]. However, this method also has been

used in estimating radiological consequences for acute releases from

severe reactor accidents [e.g., see (Pa81) , (Ri84) , and (No85)].

The coupling of the Gaussian-plume transport and deposition models

with the simple models for estimating external dose from a semi-infinite

atmospheric cloud and an infinite ground surface often results in

conservative overestimates of external dose from the finite sources.

There is one particular situation, however, where this method may result

in considerable underestimates of external dose. For exposure locations

near the point of an elevated release, the ground-level air

concentrations estimated by the Gaussian plume model may be essentially

zero, in which case the external dose based on the ground-level

concentrations also would be zero. However, the actual external dose

from the elevated plume could be considerably greater than zero due to

the ability of high-energy photons to travel large distances in air.

The photon mean-free-path in air ( i . e . , the reciprocal of the linear

attenuation coefficient) is greater than 100 m for energies above 1 MeV

(USHEW70), and appreciable external dose can occur at distances from an

extended source of several mean-free-paths.

More sophisticated calculations of external dose from a Gaussian

plume that take into account the finite dimensions of the source regions

in the air and on the ground surface necessarily involve numerical or

graphical solutions of the point kernel equation; i . e . , the integrations

over the source regions cannot be performed in closed form. It is not

our purpose to present calculations of external dose from a Gaussian

18

pi Time, but to point out that a seemingly bewildering array of approaches

to estimating external dose have been presented in the l i terature [e.g.,

see (Br74), (Br7 8), (C173) . (C066), (Ha76) , (He6 8). (He84) , (Im70),

(La81). (La82). (Mi86), (Ni85), (Ov83), (Ro79). (Sc82), and (Sh84)].

These papers consider only the calculation of external photon dose frpm

an atmospheric plume or ground deposition. Calculation of external

electron dose can be performed using the semi—infinite oloud and

infinite ground—plane approximations described above, because the range

of electrons in air generally is less than 10-20 m (NAS64).

The choice of an appropriate method for estimating external dose

from a Gaussian plume depends on a number of factors related to the type

of problem being investigated and the approximations involved in the

methods. The following discussion summarizes some of the factors that

should be considered in selecting a method.

- The formulation of the Gaussian-plume transport model for

calculating radicnuclide concentrations generally is different for

application to acute or chronic releases. For an acute release,

the plume usually is assumed to be transported away from the source

in a straight line, and further simplifying assumptions may be made

concerning the isotropy of the plume in the downwind, crosrwind,

and vertical directions. For chronic releases, however, the

calculations generally take changes in wind direction during the

release into account by averaging the plume concentrations in each

sector (an angular interval of 22.5° about the source point) over

the crosswind dimension, and crosswind averaging may be

inappropriate for estimating dose from acute releases.

- For acute releases, some methods provide dose estimates only alongthe centerline of the plume, while others allow estimation of doseat any crosswind distance relative to the centerline.

- Although most methods are appropriate for releases at any height,some may be used only for ground-level releases.

- In estimating dose in a given sector, some methods consider only

the contribution from releases to that sector, while other methods

consider the contributions from adjacent sectors or all other

sectors. This can be an important consideration for chronic

releases which are somewhat uniformly distributed in all directions

and for receptor locations relatively close to the source.

19

- Most methods use a simple approximation for the buildup factor ina i r tha t may provide considerable overestimates of dose, i . e . , thel i n e a r form (Tr66). Relat ively few methods use more exactapproximations for the buildup factor .

- With few exceptions, ca lcula t ions of external dose tha t involvenumerical in tegrat ions over a f i n i t e Gaussian plume do not takein to account plume deple t ion due to deposi t ion in es t imat ing a i rconcentrat ions as a function of locat ion. Thus, these calculaciunss t r i c t l y apply only to noble-gas radionucl ides , but provideconservative overestimates of dose for depositing radionucl ides.

Thus, i t i s clear that the use of any method for estimating externaldose from a Gaussian plume requires a c lear understanding of theassumptions used in the ca lcu la t ions and the specif ic condit ions towhich the method i s appl icable .

Other Source Configurations

The point kernel method also has been applied to a number of sourceconfigurat ions that generally are of greates t in te res t in reac tor andsource shielding appl ica t ions [ e . g . , see (B16 8), (Di7 5), (Ko68a),(Ko6Sb), (Go68a), and (Go6Sb)]. The pa r t i cu l a r source configurationstha t have been considered, and that have not been discussed previouslyin t h i s paper, include f i n i t e l ine sources, disk sources, rectangularsources, cy l indr ica l surface sources including sources containing anabsorbing medium, semi-inf i c l t e volume sources with exponential andl inear source d i s t r i bu t i ons , inf ini te slab sources with exponential andl inea r source d i s t r i bu t i ons , cyl indrical volume sources with non-absorbing or self-absorbing source medium, truncated r i g h t - c i r c u l a r conesources, and spherical volume sources with non-absorbing or self-absorbing source medium. For most of these source configurat ions, thepoint kernel method generally is amenable only to numerical solut ions ,many of which also have been presented in graphical form.

EXPOSURE SITUATIONS TO WHICH THE POINTKERNEL METHOD DOES NOT APPLY

In the second sec t ion of t h i s paper, we emphasized t h a t the pointkernel method may be regarded as a " l i n e - o f - s i g h t " method, because i tdepends only on the composition of absorbing ma te r i a l s along the d i rec t

20

path between the source /.LI receptor locations. This view is somewhat

of an oversimplification. .<ecause the photon buildup factor and electron

scaled pen.it kernel used ii the point kernel method do take into account

the scattering of emitted radiations away from the direct path and

subsequent re-scattering back to the receptor location, and they also

take into account the radiations that are emitted in directions away

from the receptor but eventually are scattered to that location.

However, the essential aspect of the method i s that the composition of

materials along any significant path of the emitted and scattered

radiations should be nearly the same as the composition along the direct

path; i.e. , the medium should be homogeneous.

There clearly are many exposure situations for which the conditions

described above are not fulfilled and, thus, for which the point kernel

method may not be appropriate. Here we need consider only external

photon exposures, again because of the relatively short range of

electrons in matter. A typical example would be a source on the ground

which is shielded in the horizontal direction but not vertically, e.g.,

a source inside a cylindrical container with no lid. In this case, an

exposure location next to the source would be shielded from photons

emitted along the direct path but not from those emitted in the vertical

direction that can re-scatter to the receptor location. A similar

example involves radioactivity buried in soil over a finite area but

near the ground surface, with the exposure location being outside the

boundary of the contaminated region. In this case, the direct paths

from all sources to the receptor location would pass through large

thicknesses of soil, and the point kernel method thus would predict

relatively low dose rates at the exposure location. However, the

calculations would not take into account that some photons emitted

vertically upward through a small thickness of soil to the atmosphere

then could be scattered in a i r to the receptor location.

A number of calcul ational methods have been developed to handle the

more complex exposure situations that involve mul ti pie. paths from source

to receptor locations that have different compositions of shielding

materials. These methods may be referred to as Monte Carlo methods

[e.g., see (Li79)], polynomial methods [e.g. , see (Be67)], moments

methods [e.g. , see (Go59)], or discrete-ordinates methods [e .g. , see

(My<5 8) ] . All of these methods involve complex numerical solutions of

radiation transport equations (as do the similar numerical methods used

to calculate buildup factors in homogeneous media for use in the point

kernel method). The essential feature of these methods that

distinguishes them from the point kernel method is that they keep track

of the energy and direction of all emitted radiations along all possible

21

paths from source to receptor locations. Thus, in the examples

described above, these methods ucn distinguish betrc-on those photons

which are emitted toward the recntor location, aur! ias are effectively

shielded before reaching that location, and those photons which are

emitted vertically upward and then re-scattered to the receptor

location. For each possible path of emitted photons, the calculations

Sake into account the probability per unit path length that a scattering

or other interaction in absorbing material will take place, the angle at

which the photon will be scattered, its energy after scattering, and

similarly to the next scattering location until the calculation is

completed for all directions of emitted photons.

Finally, we would note that the more complex methods also have been

applied to exposure situations to which the point kernel method usually

is regarded as valid. These situations include exposures to a finite

Gaussian plume in the atmosphere [e.g., see (Go82)] and exposures to

photon sources on the ground surface or distributed in soil [e.g. , see

(Be6Sb), (Be80), (Ja86a) , and (Ja86b)]. The cases of sources on the

ground or in soil provide particularly interesting comparisons with the

point kernel method, because different paths -from the source locations

to the receptor location above ground have different distances of travel

in soil. The agreement between the point kernel method and the more

sophisticated transport calculations is reasonably good for high-energy

photons (above about 0.5 MeV), but discrepancies of about 50% or more

are found as the energy decreases.

CONCLUSION

The primary purpose of t h i s paper has been to discuss the point

kernel nethod for es t imat ing ex te rna l dose from sources of photons or

e l e c t r o n s . The point kernel method i s the only approach to c a l c u l a t i n g

external dose that is amenable to relatively simple calculations.

For photons, the point kernel can be expressed as an equation,

because the specific absorbed fraction also can be expressed in

analytical form; and, for particularly simple and idealized source

geometries, the point kernel can be integrated over the source

configuration either in closed form or in terms of well known

exponential integrals. For electrons, however, general use of the point

kernel nethod requires numerical integrations of the scaled point

kernel, which is available in tabulations.

22

In environmental dose assessments, the point kernel method has been

applied to i»»alixed exposure situations invol\ ing immersion in a semi-

infinite acme pheric clond, immersion in an inf ' i i te water meditnn,

exposure to au. infinite ground surface or slab iu soil, and exposure to

contamination on the tody surface. In each case, tabulations have been

developed which give the external dose rate from photons or electrons

per unit concentration of radionuclides, and these dose-rate conversion

factors can be used directly in estimating external dose if the

concentration of radionucl ides is known. Unf ortunatelyi however, more

realistic exposure situations which commonly occur in the environment,

such as exposures to Gaussian plumes in the atmosphere or on the ground

surface, are not amenable to straightforward calculations of external

photon dose if consideration of the finite extent and spatial dependence

of the source strength are properly taken into account. In spite of the

great interest in this problem, general methods or results which can

easily be understood by the non-expert are not in widespread use.

The point kernel method also has been applied to a large number of

finite source configurations that might be found in the workplace.

However, the solutions of the method in most cases are obtainable only

with numerical integrations, and the integrations are relatively

difficult for some source configurations. Another problem in estimating

external photon dose in the workplace is that the medium surrounding a

source often is quite inhomogeneous, in which case the point kernel

method itself may not be appropriate. If the point kernel method does

not apply, then more sophisticated numerical methods for solving the

radiation transport equations are needed, but these methods generally

can be used successfully only by experts.

It is the author's experience that i t is difficult to be a casualpractitioner in the field of external dose calculations. The best thatcan be expected of the non-expert is to have a reasonable understandingof the concepts involved in estimating dose using the point kernelmethod, to know how to use dose-rate conversion factors that have beentabulated in the literature for simple and idealized exposuresituations, and to recognize when the point kernel method is not likelyto be applicable to a particular problem of interest. Beyond that, helpfrom experts invariably is required.

23

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